How do I find out if a remote system is still alive, if certain services are running, which processes are running, etc.?

You can use ping to see if the system is alive:

ping -c4 <system's IP address>

To see if a certain service is still alive, use the telnet command:

telnet <system's IP address> 25 (for SMTP)

telnet <system's IP address> 80 (for HTTP)

telnet <system's IP address> 110 (for POP3)

You can login to the system using SSH (port 22) (use PuTTY if you are on a Windows PC; PuTTY is an SSH client for Windows), and when you are on the system, you have a few useful tools to gather more information:

uptime (shows how long the system has been up)

ps aux (shows a list with all running processes)

netstat -tap (shows a list of all services that are listening on some port)

top (shows the system load)

dmesg (shows bootup messages of the last boot)

You can also have a look into the log files in /var/log/ to find out about errors, etc.